SPFirst-L07

# SPFirst-L07 - Signal Processing First Lecture 7 Fourier Series& Spectrum EE-2025 Spring-2005 jMc 3 READING ASSIGNMENTS Â This Lecture Â Fourier

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Unformatted text preview: Signal Processing First Lecture 7 Fourier Series & Spectrum 9/13/2006 EE-2025 Spring-2005 jMc 3 READING ASSIGNMENTS Â¡ This Lecture: Â¡ Fourier Series in Ch 3, Sects 3 Fourier Series in Ch 3, Sects 3-4, 3 4, 3-5 & 3 5 & 3-6 Â¡ Replaces pp. 62-66 in Ch 3 in DSP First Â¡ Notation: a k for Fourier Series Â¡ Other Reading: Â¡ Next Lecture: Sampling 9/13/2006 EE-2025 Spring-2005 jMc 4 LECTURE OBJECTIVES Â¡ ANALYSIS ANALYSIS via Fourier Series Â¡ For PERIODIC signals: x(t+T ) = x(t) Â¡ SPECTRUM SPECTRUM from Fourier Series Â¡ a k is Complex Amplitude for k-th Harmonic âˆ« âˆ’ = ) / 2 ( 1 ) ( T dt e t x a t T k j T k Ï€ 9/13/2006 EE-2025 Spring-2005 jMc 5 100 250 â€“100 â€“250 f (in Hz) 3 / 7 Ï€ j e 3 / 7 Ï€ j e âˆ’ 2 / 4 Ï€ j e âˆ’ 2 / 4 Ï€ j e 10 SPECTRUM DIAGRAM Â¡ Recall Complex Amplitude vs. Freq { } âˆ‘ = âˆ’ âˆ— + + = N k t f j k t f j k k k e a e a a t x 1 2 2 ) ( Ï€ Ï€ k j k k e A a Ï• 2 1 = âˆ— k a a 9/13/2006 EE-2025 Spring-2005 jMc 6 Harmonic Signal PERIOD/FREQUENCY of COMPLEX EXPONENTIAL: t f k j k k e a t x 2 ) ( Ï€ âˆ‘ âˆž âˆ’âˆž = = ( ) 1 or 2 2 f T T f = = = Ï€ Ï‰ Ï€ 9/13/2006 EE-2025 Spring-2005 jMc 7 Example ) 3 ( sin ) ( 3...
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## This note was uploaded on 03/03/2010 for the course EE 48976 taught by Professor Nayak during the Spring '10 term at USC.

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SPFirst-L07 - Signal Processing First Lecture 7 Fourier Series& Spectrum EE-2025 Spring-2005 jMc 3 READING ASSIGNMENTS Â This Lecture Â Fourier

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